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\centerline{Mathematics 136, section 4 -- AP Calculus}
\centerline{Information on Exam 1}
\centerline{September 24, 2003}
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{\it General Information}
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The first exam for the course will be given next Tuesday, September 30, 
as announced in the course syllabus.  It will cover the material 
discussed in class from the start of the semester
{\it through and including\/} the material on parametric curves 
from Lab Day 2 and Wednesday, September 24.  There
will be 4 or 5 problems, some possibly with several parts.  Some
may ask for a graph or the result of a calculation; others may ask for 
a precise definition of a term or concept we have used, or a 
short description or explanation of some phenomenon (similar
to the questions from Lab Day assignments). 
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{\it Calculators}
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As announced in the course syllabus, I will be providing a 
basic scientific numerical calculator (TI model
TI-30Xa) for you to 
use on this exam.  These calculators can compute $+,-,\times,\div$
and values of all functions we have talked about, but have no
graphing or symbolic computation features.  
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{\it Review Session}
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If there is interest, I will be happy to schedule a review session
outside of class time to help you get ready for the exam.  
I will be available any time after 3:00 pm on Monday, September 29
and also can arrange to be on campus the previous Saturday or Sunday
if that would be more convenient.  Start to 
prepare for the exam early, so that there will be time 
to address questions that might 
come up as you look at the review questions and sample exam 
below.
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{\it Suggested Practice Problems}
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From the text:  
\item{$\bullet$} Chapter 1 Review Problems: p. 48/1, 2, 6-17 (We did these on 
the first discussion; make sure you can do questions like this 
without a graphing calculator.  There are similar questions in 
sections 2,5,6 in Chapter 1 if you want to try others too.)
Also: 25, 29, 35. 

\item{$\bullet$} Chapter 2 Review Problems: p. 99/3,5,7,8 (for the 
last two of these, the 
directions mean: using the limit definition of the derivative).
Also: 13-17, 26, 31 (we didn't talk about anything like this in 
class -- see if you can figure out a method based on what 
we did talk about), 36
\item{$\bullet$} Be able to differentiate any functions like p. 159/1 - 52
given by formulas (you don't need to do all of these; just make sure
you can handle all the different functions involved, can spot which
derivative rule(s) apply, etc.  Also you might need to 
compute a derivative like one of these as part of a larger problem.)

\item{$\bullet$}  Be prepared for a question like 1 - 4 in Chapter 3, Section 8
(parametric curves).
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{\it Other sample exam questions:}
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\item{$\bullet$}  Imagine you are zooming in on the graphs of the following functions
near the origin:
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\itemitem{a)} $y = \sin(x) - \tan(x)$
\itemitem{b)} $y = e^x - 1$
\itemitem{c)} $y = {x^2\over x^2 + 1}$
\itemitem{d)} $y = {x\over x^2 + 1}$
\itemitem{e)} $y = {x \over x + 1}$
\itemitem{f)} $y = |x|$ (absolute value function).
\item{}
Which of them would eventually look the same?  Which would look 
locally linear?  For those that would look linear, find the equations 
of the lines you would see
{\it without\/} carrying out the zooming process.  ({\it Think of 
what we did in Lab day 1 to see the idea of this problem}).

{\it Sample Exam}  (The numbers next to each part are the possible
points on that part; they add up to 100.)
I. 
\item{A.} (10) Find a possible formula for the function $y(t)$ plotted
below:
\vglue 1.4truein
\item{B.} (10) Suppose you ``zoomed in'' on the graph of your function
near $t = 0$.  Would the graph look locally linear?  If so,
what would the equation of the line be?  If not, say why not.
(Use your formula from A, but don't do any actual ``zooming''.)
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II. A population of wildebeest grows in a habitat that can support 
no more than $P_1$ animals.  At time $t = 0$, the number of 
wildebeest present is  $P_0 < P_1$.  The population grows more and
more rapidly at first, but then the rate of growth decreases and
the population approaches $P_1$, without exceeding it.  
\item{A.} (10) Sketch a graph
of the population of wildebeest as a function of time
that fits this description.   For full credit your graph should
show proper slope and concavity.  
\item{B.} (10)  Give a qualitative graph of the derivative of your
population function.
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III.  
\item{A.}  (10) What is the exact mathematical definition of the 
{\it derivative\/} of a function $f(x)$ at $x = a$?  
\item{B.}  (10) Use the definition (not the ``shortcut rules'') 
to compute the derivative of 
$f(x) = 4x^2 - 2x + 2$ at a general $x$.
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IV.  Find the derivative functions of each of the following functions
defined by formulas:
\item{A.}  (5) $f(x) = 2x^3 + 6\sqrt{x} - {3\over \root 3\of x} + e^\pi$
\item{B.}  (7.5) $f(x) = {\sin(x) \over 1 + \cos(x)}$
\item{C.}  (7.5) $g(x) = \arctan(e^x - 2^x)$
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\bye
